1 How to burn a graph Anthony Bonato - PowerPoint Presentation

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How to burn a graph

Anthony BonatoRyerson University

GRASCan 2015

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GRASCan 2012, Ryerson University

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Emotions are contagious

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(

Kramer,Guillory,Hancock,14):

study of

emotional

or

social contagion

in Facebook

t

he underlying

network is an essential

factor

in-person interaction and

nonverbal cues are not necessary for the spread of the

contagion

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Modelling social influence

general framework:nodes are active or inactiveactive nodes are introduced and influence the activity of their neighbours

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Models

various models:(Kempe, J. Kleinberg, E. Tardos,03)competitive diffusion (Alon, et al, 2010)literature in graph theory:dominationfirefightingpercolation

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Memes

memes: an idea, behavior, or style that spreads from person to person within a culture

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Meme theory explained

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Quantifying meme outbreaks

meme breaks out at a node, then spreads to its neighbors over timememe also breaks out at other nodes over discrete time-stepshow long does it take for all nodes to receive the meme in the network?

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Burning number

G a connected, simple graph there are discrete time-steps or roundseach node is either burning or non-burningif a node is burning, then it remains in that state every round, choose an additional non-burning node to burnonce a node is burning in round t, in round t + 1, each of its non-burning neighbors becomes burningchosen nodes: activatorsprocess ends when all nodes are burning the burning number of a graph G, written by b(G), is the minimum number of rounds for all nodes to be burningwell-defined, as bounded above by |V(G)| (even (G)+1)

 

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Example: cliques

b(Kn) = 2

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Paths

burning sequence: (v3,v7,v9)sequence of activatorsTheorem (Bonato,Janssen,Roshanbin,14)b(Pn) =

 

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3

v

1

v

2

v

3

v

4

v

5

v

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v

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v

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v

9

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Proof of lower bound

suppose (x1,…,xk) is a burning sequence for Pnthen: Nk-1[x1] Nk-2[x2] N0[xk] = V(G) (1)as |Ni(x)| ≤ 2i for all nodes x, we have by (1) that: + k = 2k(k-1)/2 + k = k2 ≥ n

 

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Trees

rooted tree partition of G:

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collection

of rooted trees which are subgraphsof G, with the property that the node sets of the trees partition V(G)

x

1

, x

2

, x

3

are activators

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Trees

Theorem (BJR,14)b(G) ≤ k iff there is a rooted tree partition with trees T1,T2,…,Tk of height at most k-1, k-2, …,0 (respectively)such that for all i, j, the roots of Ti and Tj are distance at least |i-j|.

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Trees

note: if H is a spanning subgraph of G, then b(G) ≤ b(H)a burning sequence for H is also one for GCorollary (BJR,14)b(G) = min{b(T): T is a spanning tree of G}

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Bounds

Corollary (BJR,14)b(Cn) =If G has a Hamiltonian path, then b(G) ≤ burning is not monotonic in general on subgraphs; even for isometric subgraphseg b(C5) = 3 > b(W5) = 2Lemma (BJR,14) If H is an isometric subtree of G, then b(H) ≤ b(G).

 

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Aside: spider graphs

SP(3,5):Lemma (BJR,14) b(SP(s,r)) = r+1.

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Bounds

Theorem (BJR,14)If G has diameter d and radius r, then≤ b(G) ≤ r+1.tight: upper bound: spider graphslower bound: paths

 

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Coverings

Theorem (BJR,14)If C1,C2,…,Ct cover G, and each Ci is connected of radius at most k, then b(G) ≤ t + k.(G): k-distance domination numberCorollary (BJR,14) k}

 

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How large can the burning number be?

Conjecture (BJR,14): b(G) ≤ .by using corollary on we have that: b(G) ≤ 2-1.(Coudert,Nisse,Roshanbin,15+): b(G) ≤

 

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Nordhaus-Gaddum type results

Theorem (BJR,15+)If n ≥ 2, then 4 ≤ b(G) + b() ≤ n+2. If G is connected and n ≥ 6, then b(G) + b() ≤ -1.Theorem (BJR,15+)If n ≥ 6, then 4 ≤ b(G)b() ≤ 2n. If G is connected, then b(G)b() ≤ n + 6.Conjecture (BJR,15+): b(G)b() ≤ n + 4.

 

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Iterated Local Transitivity (ILT) model(Bonato, Hadi, Horn, Prałat, Wang, 08)

key paradigm is

transitivity

: friends of friends are more likely friends; eg

(Girvan and Newman, 03)

iterative cloning of closed neighbour sets

deterministic

local

: nodes often only have local

influence

evolves

over

time

, but retains

memory of initial graph

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ILT model

begin with a graph

G = G

0

to form the graph

G

t+1

for each vertex

x

from time

t

, add a vertex

x’

, the

clone of

x

, so that

xx’

is an edge, and

x’

is joined to each neighbor of

x

order of

G

t

is

2

t

n

0

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G

0 = C4

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Properties of ILT model

average degree

increasing

to

∞

with time

average distance

bounded by constant and converging

, and in many cases

decreasing

with time; diameter does not change

clustering

higher

than in a random generated graph with same average degree

bad expansion

: small gaps between 1

st

and 2

nd

eigenvalues in adjacency and normalized Laplacian matrices of

G

t

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Burning ILT

although ILT generates graphs with exponential order/size, the burning number is constant:Theorem (BJR,14) For all t, b(Gt) {b(G0), b(G0)+1}.

 

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Cartesian grids

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Cartesian grids

Theorem (BJR,14)If 1 ≤ m ≤ n, and G is the m x n Cartesian grid, then we have the following:If m = O(), then b(G) = If m = ), then b(G) =

 

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Sketch of proof

consider upper bound in the case m = O() idea: using a covering by t closed balls of radius r (diamonds), with r to be determinedgives upper bound for b(G) of t+r by covering theorem

 

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2r+1

2r+1

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Sketch of proof

now let r = (Mitsche,Prałat,Roshanbin,15+) derived constantsfor the n x n grid, b(G) =

 

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Complexity

Burning number problem:Instance: A graph G and an integer k ≥ 2.Question: Is b(G) ≤ k?in NP

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Burning a graph is hard

Theorem (BJR,14+) The Burning number problem is NP-hard.Further, it is NP-hard when restricted to any one of the following graph classes:planar graphs disconnected graphsbipartite graphsreduction from planar 3-SAT

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Gadgets

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Burning a graph is hard

Theorem (BJR,15+) The Burning number problem is NP-hard when restricted to trees of maximum degree 3.reduction from a certain subset-sum problem

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Random burning

select activators at randomwe consider uniform choice with replacement

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Cost of drunkeness

bR(G): random variable associated with the first time all vertices of G are burningb(G) ≤ bR(G)C(G) = bR(G)/b(G): cost of drunkenness

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Drunkeness on paths

Theorem (Mitsche,Pralat,Roshanbin,15+)C(Pn) = first and second moment methods

 

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Other random burning models

choose activatorswithout replacementfrom non-burning verticesfor (1), cost of drunkenness on paths is unchanged, asymptoticallyfor (2), cost of drunkenness is constant

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Future directions

conjecture: b(G) ≤ burning in grids strong, hexagonal, triangulargrids?3-dimensional?infinite grids?burning in graph products Cartesian, strong, categorical

 

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Future directions

random graphs and cost of drunkennessbinomial, geometric random graphs (MPR,15+)random regular?drunkenness in hypercubes?graph bootstrap percolationvertices burn if joined to r >1 burning verticesburning in models for complex networkspreferential attachment, copying, geometric models?

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